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## Heuristic pattern search for bound constrained minimax problems

Fonte: Springer-Verlag
Publicador: Springer-Verlag

Tipo: Parte de Livro

Publicado em //2011
ENG

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This paper presents a pattern search algorithm and its hybridization
with a random descent search for solving bound constrained minimax problems.
The herein proposed heuristic pattern search method combines the Hooke and
Jeeves (HJ) pattern and exploratory moves with a randomly generated approxi-
mate descent direction. Two versions of the heuristic algorithm have been applied
to several benchmark minimax problems and compared with the original HJ pat-
tern search algorithm.

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## Seleção da carteira de ativos de uma seguradora em tempos de crise : o critério Minimax

Fonte: Instituto Superior de Economia e Gestão
Publicador: Instituto Superior de Economia e Gestão

Tipo: Dissertação de Mestrado

Publicado em /09/2012
POR

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Mestrado em Decisão Económica e Empresarial; A atual crise económica e financeira trouxe novos contornos ao problema da seleção de carteiras de ativos. No caso particular das companhias seguradoras, tradicionalmente pautadas por critérios prudenciais, a questão assume importância acrescida.
Neste trabalho, com base em Polak et al. (2010), apresenta-se a aplicação dos modelos Minimax na escolha da carteira ótima de uma seguradora, de modo a obter uma rentabilidade mínima, qualquer que seja o cenário razoavelmente previsível. São resolvidos três problemas, distintos mas interrelacionados, em certas condições.
Fazem-se duas extensões ao modelo original: a introdução de um conjunto muito mais alargado de estados da natureza, integrando de modo explícito os cenários de crise; a modelização das séries temporais das rentabilidades dos ativos elegíveis, para fins de previsão dos estados da natureza futuros. Em ambos os casos se obtêm resultados muito satisfatórios.; The present economic and financial crisis brought new issues to the asset portfolio selection problem. As insurance companies are traditionally bound to prudential criteria, the topic is particularly important in their context. In this work, inspired in Polak et al. (2010)...

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## Convex Sets Strict Separation in the Minimax Theorem

Fonte: Hikari Ltd.
Publicador: Hikari Ltd.

Tipo: Artigo de Revista Científica

Publicado em //2014
ENG

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The convex sets strict separation is very useful to obtain mathematical optimization
results. The minimax theorem, a key result in Game Theory is an example. It will be outlined in this work.

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## Hierarchical Clustering With Prototypes via Minimax Linkage

Fonte: PubMed
Publicador: PubMed

Tipo: Artigo de Revista Científica

Publicado em //2011
EN

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Agglomerative hierarchical clustering is a popular class of methods for understanding the structure of a dataset. The nature of the clustering depends on the choice of linkage—that is, on how one measures the distance between clusters. In this article we investigate minimax linkage, a recently introduced but little-studied linkage. Minimax linkage is unique in naturally associating a prototype chosen from the original dataset with every interior node of the dendrogram. These prototypes can be used to greatly enhance the interpretability of a hierarchical clustering. Furthermore, we prove that minimax linkage has a number of desirable theoretical properties; for example, minimax-linkage dendrograms cannot have inversions (unlike centroid linkage) and is robust against certain perturbations of a dataset. We provide an efficient implementation and illustrate minimax linkage’s strengths as a data analysis and visualization tool on a study of words from encyclopedia articles and on a dataset of images of human faces.

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## Tuning support vector machines for minimax and Neyman-Pearson classification

Fonte: Universidade Rice
Publicador: Universidade Rice

Tipo: Relatório

EN_US

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#error estimation#minimax classification#support vector machines#Neyman-Pearson classification#parameter selection

This paper studies the training of support vector machine (SVM) classifiers with respect to the minimax and Neyman-Pearson criteria. In principle, these criteria can be optimized in a straightforward way using a cost-sensitive SVM. In practice, however, because these criteria require especially accurate error estimation, standard techniques for tuning SVM parameters, such as crossvalidation, can lead to poor classifier performance. To address this issue, we first prove that the usual cost-sensitive SVM, here called the 2C-SVM, is equivalent to another formulation called the 2nu-SVM. We then exploit a characterization of the 2nu-SVM parameter space to develop a simple yet powerful approach to error estimation based on smoothing. In an extensive experimental study we demonstrate that smoothing significantly improves the accuracy of cross-validation error estimates, leading to dramatic performance gains. Furthermore, we propose coordinate descent strategies that offer significant gains in computational efficiency, with little to no loss in performance.

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## Prediction and Filtering of Stationary Processes: Yaglom’s Method and Minimax Filtering

Fonte: Quens University
Publicador: Quens University

Tipo: Tese de Doutorado

EN

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The aim of this work is to give a basic introduction to the theory of stationary stochastic processes, particularly to the somewhat specialized problem of prediction and filtering of such processes. Kolmogorov was the first to make
a contribution to its solution using involved mathematical theory. In the years following the publication of Wiener’s famous book, the theory gained considerable popularity from the applied sciences, particularly radio engineering. In this work, we shall present Yaglom’s method to solving the problems considered in Wiener’s book. This alternative approach is entirely based on rather basic facts from Hilbert space theory and the theory of complex variables. As it turns out, the theory of filtering of stationary processes heavily relies on spectral properties of the processes. In particular, Yaglom’s approach assumes complete knowledge of the spectral densities. In this work, however, we shall not be concerned with the problem of estimating such quantities based on a finite sample. Instead, in order to account for uncertainty as frequently encountered in practice, we shall discuss the problem of minimax filtering which has emerged from the practical need of allowing for incomplete knowledge about spectral properties.

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## Teoria dos jogos e a relação entre o “Teorema Minimax” de John Von Neumann e o “Equilíbrio de Nash” de John Nash

Fonte: Universidade Católica de Brasília
Publicador: Universidade Católica de Brasília

Tipo: Trabalho de Conclusão de Curso
Formato: Texto

PT_BR

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Este texto contém um pouco da Teoria dos Jogos baseada nos trabalhos de John Von Neumann e de John Forbes
Nash Jr, aplicada a problemas que podem ser enfrentados em situações do mundo real. A estrutura do texto
segue uma ordem de construção do conhecimento sobre a teoria, iniciando pela idéia geral, passando pelo
Teorema Minimax e finalizando com o Equilíbrio de Nash; Matemática

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## Empirical Entropy, Minimax Regret and Minimax Risk

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/08/2013

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We consider the random design regression model with square loss. We propose a
method that aggregates empirical minimizers (ERM) over appropriately chosen
random subsets and reduces to ERM in the extreme case, and we establish sharp
oracle inequalities for its risk. We show that, under the $\epsilon^{-p}$
growth of the empirical $\epsilon$-entropy, the excess risk of the proposed
method attains the rate $n^{-\frac{2}{2+p}}$ for $p\in(0,2]$ and $n^{-1/p}$ for
$p> 2$ where $n$ is the sample size. Furthermore, for $p\in(0,2]$, the excess
risk rate matches the behavior of the minimax risk of function estimation in
regression problems under the well-specified model. This yields a conclusion
that the rates of statistical estimation in well-specified models (minimax
risk) and in misspecified models (minimax regret) are equivalent in the regime
$p\in(0,2]$. In other words, for $p\in(0,2]$ the problem of statistical
learning enjoys the same minimax rate as the problem of statistical estimation.
On the contrary, for $p>2$ we show that the rates of the minimax regret are, in
general, slower than for the minimax risk. Our oracle inequalities also imply
the $v\log(n/v)/n$ rates for Vapnik-Chervonenkis type classes of dimension $v$
without the usual convexity assumption on the class; we show that these rates
are optimal. Finally...

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## Minimax state estimation for linear descriptor systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/02/2011

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Author's Summary of the dissertation for the degree of the Candidate of
Science (physics and mathematics). The aim of the dissertation is to develop a
generalized Kalman Duality concept applicable for linear unbounded
non-invertible operators and introduce the minimax state estimation theory and
algorithms for linear differential-algebraic equations. In particular, the
dissertation pursues the following goals: - develop generalized duality concept
for the minimax state estimation theory for DAEs with unknown but bounded model
error and random observation noise with unknown but bounded correlation
operator; - derive the minimax state estimation theory for linear DAEs with
unknown but bounded model error and random observation noise with unknown but
bounded correlation operator; - describe how the DAE model propagates uncertain
parameters; - estimate the worst-case error; - construct fast estimation
algorithms in the form of filters; - develop a tool for model validation, that
is to assess how good the model describes observed phenomena.
The dissertation contains the following new results: - generalized version of
the Kalman duality principle is proposed allowing to handle unbounded linear
model operators with non-trivial null-space; - new definitions of the minimax
estimates for DAEs based on the generalized Kalman duality principle are
proposed; - theorems of existence for minimax estimates are proved; - new
minimax state estimation algorithms (in the form of filter and in the
variational form) for DAE are proposed.

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## Optimal Grouping for Group Minimax Hypothesis Testing

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/07/2013

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Bayesian hypothesis testing and minimax hypothesis testing represent extreme
instances of detection in which the prior probabilities of the hypotheses are
either completely and precisely known, or are completely unknown. Group
minimax, also known as Gamma-minimax, is a robust intermediary between Bayesian
and minimax hypothesis testing that allows for coarse or partial advance
knowledge of the hypothesis priors by using information on sets in which the
prior lies. Existing work on group minimax, however, does not consider the
question of how to define the sets or groups of priors; it is assumed that the
groups are given. In this work, we propose a novel intermediate detection
scheme formulated through the quantization of the space of prior probabilities
that optimally determines groups and also representative priors within the
groups. We show that when viewed from a quantization perspective, group minimax
amounts to determining centroids with a minimax Bayes risk error divergence
distortion criterion: the appropriate Bregman divergence for this task.
Moreover, the optimal partitioning of the space of prior probabilities is a
Bregman Voronoi diagram. Together, the optimal grouping and representation
points are an epsilon-net with respect to Bayes risk error divergence...

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## Torsion-free covers for solvable minimax groups

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/10/2015

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We prove that every finitely generated solvable minimax group can be realized
as a quotient of a torsion-free solvable minimax group. This result has an
application to the investigation of random walks on finitely generated solvable
minimax groups. Our methods also allow us to completely characterize the
solvable minimax groups that are homomorphic images of torsion-free solvable
minimax groups.

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## Improved minimax estimation of a multivariate normal mean under heteroscedasticity

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/05/2015

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Consider the problem of estimating a multivariate normal mean with a known
variance matrix, which is not necessarily proportional to the identity matrix.
The coordinates are shrunk directly in proportion to their variances in Efron
and Morris' (J. Amer. Statist. Assoc. 68 (1973) 117-130) empirical Bayes
approach, whereas inversely in proportion to their variances in Berger's (Ann.
Statist. 4 (1976) 223-226) minimax estimators. We propose a new minimax
estimator, by approximately minimizing the Bayes risk with a normal prior among
a class of minimax estimators where the shrinkage direction is open to
specification and the shrinkage magnitude is determined to achieve minimaxity.
The proposed estimator has an interesting simple form such that one group of
coordinates are shrunk in the direction of Berger's estimator and the remaining
coordinates are shrunk in the direction of the Bayes rule. Moreover, the
proposed estimator is scale adaptive: it can achieve close to the minimum Bayes
risk simultaneously over a scale class of normal priors (including the
specified prior) and achieve close to the minimax linear risk over a
corresponding scale class of hyper-rectangles. For various scenarios in our
numerical study, the proposed estimators with extreme priors yield more
substantial risk reduction than existing minimax estimators.; Comment: Published at http://dx.doi.org/10.3150/13-BEJ580 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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## Minimax Estimation of Nonregular Parameters and Discontinuity in Minimax Risk

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/03/2014

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When a parameter of interest is nondifferentiable in the probability, the
existing theory of semiparametric efficient estimation is not applicable, as it
does not have an influence function. Song (2014) recently developed a local
asymptotic minimax estimation theory for a parameter that is a
nondifferentiable transform of a regular parameter, where the nondifferentiable
transform is a composite map of a continuous piecewise linear map with a single
kink point and a translation-scale equivariant map. The contribution of this
paper is two fold. First, this paper extends the local asymptotic minimax
theory to nondifferentiable transforms that are a composite map of a Lipschitz
continuous map having a finite set of nondifferentiability points and a
translation-scale equivariant map. Second, this paper investigates the
discontinuity of the local asymptotic minimax risk in the true probability and
shows that the proposed estimator remains to be optimal even when the risk is
locally robustified not only over the scores at the true probability, but also
over the true probability itself. However, the local robustification does not
resolve the issue of discontinuity in the local asymptotic minimax risk.

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## Minimax Robust Function Reconstruction in Reproducing Kernel Hilbert Spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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In this paper, we present a unified approach to function approximation in
reproducing kernel Hilbert spaces (RKHS) that establishes a previously
unrecognized optimality property for several well-known function approximation
techniques, such as minimum-norm interpolation, smoothing splines, and
pseudo-inverses. We consider the problem of approximating a function belonging
to an arbitrary real-valued RKHS on R^d based on approximate observations of
the function. The observations are approximate in the sense that the actual
observations (i.e., the true function values) are known only to belong to a
convex set of admissible observations. We seek a minimax optimal approximation
for the function that minimizes the supremum of the RKHS norm on the error
between the true function and the chosen approximation subject only to the
conditions that the true function belongs to a uniformly bounded uncertainty
set of functions that satisfy the constraints on the observations and that the
approximation is a member of the RKHS. We refer to such a solution as a minimax
robust reconstruction. We characterize the solution to the minimax robust
reconstruction problem and show that it is equivalent to solving a
straightforward convex optimization problem. We demonstrate that a minimax
robust reconstruction will generally be more stable than an approximation based
on interpolation through a nominal set of observations and that...

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## Mini-Minimax Uncertainty Quantification for Emulators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Statistics - Methodology#Statistics - Applications#Statistics - Computation#68Q17, 65D05, 68U20, 62P12

Consider approximating a "black box" function $f$ by an emulator $\hat{f}$
based on $n$ noiseless observations of $f$. Let $w$ be a point in the domain of
$f$. How big might the error $|\hat{f}(w) - f(w)|$ be? If $f$ could be
arbitrarily rough, this error could be arbitrarily large: we need some
constraint on $f$ besides the data. Suppose $f$ is Lipschitz with known
constant. We find a lower bound on the number of observations required to
ensure that for the best emulator $\hat{f}$ based on the $n$ data, $|\hat{f}(w)
- f(w)| \le \epsilon$. But in general, we will not know whether $f$ is
Lipschitz, much less know its Lipschitz constant. Assume optimistically that
$f$ is Lipschitz-continuous with the smallest constant consistent with the $n$
data. We find the maximum (over such regular $f$) of $|\hat{f}(w) - f(w)|$ for
the best possible emulator $\hat{f}$; we call this the "mini-minimax
uncertainty" at $w$. In reality, $f$ might not be Lipschitz or---if it is---it
might not attain its Lipschitz constant on the data. Hence, the mini-minimax
uncertainty at $w$ could be much smaller than $|\hat{f}(w) - f(w)|$. But if the
mini-minimax uncertainty is large, then---even if $f$ satisfies the optimistic
regularity assumption---$|\hat{f}(w) - f(w)|$ could be large...

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## Minimax Filtering via Relations between Information and Estimation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We investigate the problem of continuous-time causal estimation under a
minimax criterion. Let $X^T = \{X_t,0\leq t\leq T\}$ be governed by the
probability law $P_{\theta}$ from a class of possible laws indexed by $\theta
\in \Lambda$, and $Y^T$ be the noise corrupted observations of $X^T$ available
to the estimator. We characterize the estimator minimizing the worst case
regret, where regret is the difference between the causal estimation loss of
the estimator and that of the optimum estimator.
One of the main contributions of this paper is characterizing the minimax
estimator, showing that it is in fact a Bayesian estimator. We then relate
minimax regret to the channel capacity when the channel is either Gaussian or
Poisson. In this case, we characterize the minimax regret and the minimax
estimator more explicitly. If we further assume that the uncertainty set
consists of deterministic signals, the worst case regret is exactly equal to
the corresponding channel capacity, namely the maximal mutual information
attainable across the channel among all possible distributions on the
uncertainty set of signals. The corresponding minimax estimator is the Bayesian
estimator assuming the capacity-achieving prior. Using this relation, we also
show that the capacity achieving prior coincides with the least favorable
input. Moreover...

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## Predicting The Performance of Minimax and Product in Game-Tree

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/03/2013

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The discovery that the minimax decision rule performs poorly in some games
has sparked interest in possible alternatives to minimax. Until recently, the
only games in which minimax was known to perform poorly were games which were
mainly of theoretical interest. However, this paper reports results showing
poor performance of minimax in a more common game called kalah. For the kalah
games tested, a non-minimax decision rule called the product rule performs
significantly better than minimax.
This paper also discusses a possible way to predict whether or not minimax
will perform well in a game when compared to product. A parameter called the
rate of heuristic flaw (rhf) has been found to correlate positively with the.
performance of product against minimax. Both analytical and experimental
results are given that appear to support the predictive power of rhf.; Comment: Appears in Proceedings of the Second Conference on Uncertainty in
Artificial Intelligence (UAI1986)

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## Minimax Multi-Task Learning and a Generalized Loss-Compositional Paradigm for MTL

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 13/09/2012

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Since its inception, the modus operandi of multi-task learning (MTL) has been
to minimize the task-wise mean of the empirical risks. We introduce a
generalized loss-compositional paradigm for MTL that includes a spectrum of
formulations as a subfamily. One endpoint of this spectrum is minimax MTL: a
new MTL formulation that minimizes the maximum of the tasks' empirical risks.
Via a certain relaxation of minimax MTL, we obtain a continuum of MTL
formulations spanning minimax MTL and classical MTL. The full paradigm itself
is loss-compositional, operating on the vector of empirical risks. It
incorporates minimax MTL, its relaxations, and many new MTL formulations as
special cases. We show theoretically that minimax MTL tends to avoid worst case
outcomes on newly drawn test tasks in the learning to learn (LTL) test setting.
The results of several MTL formulations on synthetic and real problems in the
MTL and LTL test settings are encouraging.; Comment: appearing at NIPS 2012

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## Nonparametric Regression Estimation Based on Spatially Inhomogeneous Data: Minimax Global Convergence Rates and Adaptivity

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We consider the nonparametric regression estimation problem of recovering an
unknown response function f on the basis of spatially inhomogeneous data when
the design points follow a known compactly supported density g with a finite
number of well separated zeros. In particular, we consider two different cases:
when g has zeros of a polynomial order and when g has zeros of an exponential
order. These two cases correspond to moderate and severe data losses,
respectively. We obtain asymptotic minimax lower bounds for the global risk of
an estimator of f and construct adaptive wavelet nonlinear thresholding
estimators of f which attain those minimax convergence rates (up to a
logarithmic factor in the case of a zero of a polynomial order), over a wide
range of Besov balls.
The spatially inhomogeneous ill-posed problem that we investigate is
inherently more difficult than spatially homogeneous problems like, e.g.,
deconvolution. In particular, due to spatial irregularity, assessment of
minimax global convergence rates is a much harder task than the derivation of
minimax local convergence rates studied recently in the literature.
Furthermore, the resulting estimators exhibit very different behavior and
minimax global convergence rates in comparison with the solution of spatially
homogeneous ill-posed problems. For example...

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## A Minimax Robust Decoding Algorithm

Fonte: Institute of Electrical and Electronics Engineers (IEEE Inc)
Publicador: Institute of Electrical and Electronics Engineers (IEEE Inc)

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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#Keywords: Decoding algorithm#Impulsive noise#Sum product algorithm#Turbo decoding#Viterbi algorithm#Algorithms#Communication channels (information theory)#Optimization#Probability density function#Signal processing#Spurious signal noise

In this correspondence we study the decoding problem in an uncertain noise environment. If the receiver knows the noise probability density function (pdf) at each time slot or its a priori probability, the standard Viterbi algorithm (VA) or the a posteriori probability (APP) algorithm can achieve optimal performance. However, if the actual noise distribution differs from the noise model used to design the receiver, there can be significant performance degradation due to the model mismatch. The minimax concept is used to minimize the worst possible error performance over a family of possible channel noise pdf's. We show that the optimal robust scheme is difficult to derive; therefore, alternative, practically feasible, robust decoding schemes are presented and implemented on VA decoder and two-way APP decoder. Performance analysis and numerical results show our robust decoders have a performance advantage over standard decoders in uncertain noise channels, with no or little computational overhead. Our robust decoding approach can also explain why for turbo decoding overestimating the noise variance gives better results than underestimating it.

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